The rewrite relation of the following TRS is considered.
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(2ndspos(0,Z)) |
→ |
mark(rnil) |
(2) |
active(2ndspos(s(N),cons(X,cons(Y,Z)))) |
→ |
mark(rcons(posrecip(Y),2ndsneg(N,Z))) |
(3) |
active(2ndsneg(0,Z)) |
→ |
mark(rnil) |
(4) |
active(2ndsneg(s(N),cons(X,cons(Y,Z)))) |
→ |
mark(rcons(negrecip(Y),2ndspos(N,Z))) |
(5) |
active(pi(X)) |
→ |
mark(2ndspos(X,from(0))) |
(6) |
active(plus(0,Y)) |
→ |
mark(Y) |
(7) |
active(plus(s(X),Y)) |
→ |
mark(s(plus(X,Y))) |
(8) |
active(times(0,Y)) |
→ |
mark(0) |
(9) |
active(times(s(X),Y)) |
→ |
mark(plus(Y,times(X,Y))) |
(10) |
active(square(X)) |
→ |
mark(times(X,X)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(12) |
active(posrecip(X)) |
→ |
posrecip(active(X)) |
(13) |
active(negrecip(X)) |
→ |
negrecip(active(X)) |
(14) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(15) |
active(rcons(X1,X2)) |
→ |
rcons(active(X1),X2) |
(16) |
active(rcons(X1,X2)) |
→ |
rcons(X1,active(X2)) |
(17) |
active(from(X)) |
→ |
from(active(X)) |
(18) |
active(2ndspos(X1,X2)) |
→ |
2ndspos(active(X1),X2) |
(19) |
active(2ndspos(X1,X2)) |
→ |
2ndspos(X1,active(X2)) |
(20) |
active(2ndsneg(X1,X2)) |
→ |
2ndsneg(active(X1),X2) |
(21) |
active(2ndsneg(X1,X2)) |
→ |
2ndsneg(X1,active(X2)) |
(22) |
active(pi(X)) |
→ |
pi(active(X)) |
(23) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(24) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(25) |
active(times(X1,X2)) |
→ |
times(active(X1),X2) |
(26) |
active(times(X1,X2)) |
→ |
times(X1,active(X2)) |
(27) |
active(square(X)) |
→ |
square(active(X)) |
(28) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
negrecip(mark(X)) |
→ |
mark(negrecip(X)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(32) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
2ndspos(mark(X1),X2) |
→ |
mark(2ndspos(X1,X2)) |
(36) |
2ndspos(X1,mark(X2)) |
→ |
mark(2ndspos(X1,X2)) |
(37) |
2ndsneg(mark(X1),X2) |
→ |
mark(2ndsneg(X1,X2)) |
(38) |
2ndsneg(X1,mark(X2)) |
→ |
mark(2ndsneg(X1,X2)) |
(39) |
pi(mark(X)) |
→ |
mark(pi(X)) |
(40) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
square(mark(X)) |
→ |
mark(square(X)) |
(45) |
proper(0) |
→ |
ok(0) |
(46) |
proper(s(X)) |
→ |
s(proper(X)) |
(47) |
proper(posrecip(X)) |
→ |
posrecip(proper(X)) |
(48) |
proper(negrecip(X)) |
→ |
negrecip(proper(X)) |
(49) |
proper(nil) |
→ |
ok(nil) |
(50) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(51) |
proper(rnil) |
→ |
ok(rnil) |
(52) |
proper(rcons(X1,X2)) |
→ |
rcons(proper(X1),proper(X2)) |
(53) |
proper(from(X)) |
→ |
from(proper(X)) |
(54) |
proper(2ndspos(X1,X2)) |
→ |
2ndspos(proper(X1),proper(X2)) |
(55) |
proper(2ndsneg(X1,X2)) |
→ |
2ndsneg(proper(X1),proper(X2)) |
(56) |
proper(pi(X)) |
→ |
pi(proper(X)) |
(57) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(58) |
proper(times(X1,X2)) |
→ |
times(proper(X1),proper(X2)) |
(59) |
proper(square(X)) |
→ |
square(proper(X)) |
(60) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
negrecip(ok(X)) |
→ |
ok(negrecip(X)) |
(63) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(64) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
2ndspos(ok(X1),ok(X2)) |
→ |
ok(2ndspos(X1,X2)) |
(67) |
2ndsneg(ok(X1),ok(X2)) |
→ |
ok(2ndsneg(X1,X2)) |
(68) |
pi(ok(X)) |
→ |
ok(pi(X)) |
(69) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
square(ok(X)) |
→ |
ok(square(X)) |
(72) |
top(mark(X)) |
→ |
top(proper(X)) |
(73) |
top(ok(X)) |
→ |
top(active(X)) |
(74) |
There are 113 ruless (increase limit for explicit display).
The dependency pairs are split into 15
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(187) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(179) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(cons#) |
= |
1 |
π(top#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(posrecip#) |
= |
1 |
π(active) |
= |
1 |
π(posrecip) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(negrecip) |
= |
2 |
|
status(negrecip) |
= |
[1] |
|
list-extension(negrecip) |
= |
Lex |
prec(s) |
= |
1 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(negrecip#) |
= |
0 |
|
status(negrecip#) |
= |
[] |
|
list-extension(negrecip#) |
= |
Lex |
prec(2ndspos) |
= |
3 |
|
status(2ndspos) |
= |
[1, 2] |
|
list-extension(2ndspos) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(rnil) |
= |
3 |
|
status(rnil) |
= |
[] |
|
list-extension(rnil) |
= |
Lex |
prec(plus#) |
= |
0 |
|
status(plus#) |
= |
[1, 2] |
|
list-extension(plus#) |
= |
Lex |
prec(square) |
= |
2 |
|
status(square) |
= |
[1] |
|
list-extension(square) |
= |
Lex |
prec(square#) |
= |
0 |
|
status(square#) |
= |
[] |
|
list-extension(square#) |
= |
Lex |
prec(pi) |
= |
4 |
|
status(pi) |
= |
[1] |
|
list-extension(pi) |
= |
Lex |
prec(rcons#) |
= |
0 |
|
status(rcons#) |
= |
[1] |
|
list-extension(rcons#) |
= |
Lex |
prec(rcons) |
= |
2 |
|
status(rcons) |
= |
[2, 1] |
|
list-extension(rcons) |
= |
Lex |
prec(times#) |
= |
0 |
|
status(times#) |
= |
[] |
|
list-extension(times#) |
= |
Lex |
prec(0) |
= |
5 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(from) |
= |
5 |
|
status(from) |
= |
[1] |
|
list-extension(from) |
= |
Lex |
prec(times) |
= |
5 |
|
status(times) |
= |
[2, 1] |
|
list-extension(times) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(nil) |
= |
0 |
|
status(nil) |
= |
[] |
|
list-extension(nil) |
= |
Lex |
prec(mark) |
= |
1 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(2ndsneg) |
= |
3 |
|
status(2ndsneg) |
= |
[1, 2] |
|
list-extension(2ndsneg) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(plus) |
= |
3 |
|
status(plus) |
= |
[2, 1] |
|
list-extension(plus) |
= |
Lex |
prec(2ndspos#) |
= |
0 |
|
status(2ndspos#) |
= |
[2] |
|
list-extension(2ndspos#) |
= |
Lex |
prec(from#) |
= |
0 |
|
status(from#) |
= |
[] |
|
list-extension(from#) |
= |
Lex |
prec(cons) |
= |
4 |
|
status(cons) |
= |
[1] |
|
list-extension(cons) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
prec(pi#) |
= |
0 |
|
status(pi#) |
= |
[] |
|
list-extension(pi#) |
= |
Lex |
prec(2ndsneg#) |
= |
0 |
|
status(2ndsneg#) |
= |
[] |
|
list-extension(2ndsneg#) |
= |
Lex |
and the following
Max-polynomial interpretation
[negrecip(x1)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
x1 + 0 |
[negrecip#(x1)] |
=
|
1 |
[2ndspos(x1, x2)] |
=
|
x1 + x2 + 7433 |
[top(x1)] |
=
|
1 |
[rnil] |
=
|
40080 |
[plus#(x1, x2)] |
=
|
max(x1 + 1, x2 + 1, 0) |
[square(x1)] |
=
|
x1 + 15946 |
[square#(x1)] |
=
|
1 |
[pi(x1)] |
=
|
x1 + 58092 |
[rcons#(x1, x2)] |
=
|
max(x1 + 1, 0) |
[rcons(x1, x2)] |
=
|
max(x1 + 7434, x2 + 0, 0) |
[times#(x1, x2)] |
=
|
max(0) |
[0] |
=
|
32648 |
[from(x1)] |
=
|
x1 + 18010 |
[times(x1, x2)] |
=
|
max(x1 + 15945, x2 + 15944, 0) |
[s#(x1)] |
=
|
1 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[2ndsneg(x1, x2)] |
=
|
x1 + x2 + 7433 |
[proper#(x1)] |
=
|
1 |
[plus(x1, x2)] |
=
|
max(x1 + 1, x2 + 0, 0) |
[2ndspos#(x1, x2)] |
=
|
x2 + 1 |
[from#(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
max(x1 + 2749, x2 + 0, 0) |
[active#(x1)] |
=
|
1 |
[pi#(x1)] |
=
|
1 |
[2ndsneg#(x1, x2)] |
=
|
x1 + 1 |
together with the usable
rules
active(from(X)) |
→ |
from(active(X)) |
(18) |
proper(nil) |
→ |
ok(nil) |
(50) |
active(2ndsneg(0,Z)) |
→ |
mark(rnil) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(15) |
active(plus(s(X),Y)) |
→ |
mark(s(plus(X,Y))) |
(8) |
proper(from(X)) |
→ |
from(proper(X)) |
(54) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(1) |
active(2ndspos(s(N),cons(X,cons(Y,Z)))) |
→ |
mark(rcons(posrecip(Y),2ndsneg(N,Z))) |
(3) |
active(rcons(X1,X2)) |
→ |
rcons(active(X1),X2) |
(16) |
active(2ndsneg(X1,X2)) |
→ |
2ndsneg(active(X1),X2) |
(21) |
2ndspos(mark(X1),X2) |
→ |
mark(2ndspos(X1,X2)) |
(36) |
2ndsneg(ok(X1),ok(X2)) |
→ |
ok(2ndsneg(X1,X2)) |
(68) |
active(times(X1,X2)) |
→ |
times(active(X1),X2) |
(26) |
negrecip(ok(X)) |
→ |
ok(negrecip(X)) |
(63) |
active(2ndspos(X1,X2)) |
→ |
2ndspos(active(X1),X2) |
(19) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(32) |
active(rcons(X1,X2)) |
→ |
rcons(X1,active(X2)) |
(17) |
proper(square(X)) |
→ |
square(proper(X)) |
(60) |
active(times(X1,X2)) |
→ |
times(X1,active(X2)) |
(27) |
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
active(2ndsneg(X1,X2)) |
→ |
2ndsneg(X1,active(X2)) |
(22) |
active(square(X)) |
→ |
square(active(X)) |
(28) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
active(2ndsneg(s(N),cons(X,cons(Y,Z)))) |
→ |
mark(rcons(negrecip(Y),2ndspos(N,Z))) |
(5) |
square(ok(X)) |
→ |
ok(square(X)) |
(72) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(64) |
active(times(s(X),Y)) |
→ |
mark(plus(Y,times(X,Y))) |
(10) |
2ndsneg(X1,mark(X2)) |
→ |
mark(2ndsneg(X1,X2)) |
(39) |
active(plus(0,Y)) |
→ |
mark(Y) |
(7) |
active(2ndspos(X1,X2)) |
→ |
2ndspos(X1,active(X2)) |
(20) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(25) |
proper(negrecip(X)) |
→ |
negrecip(proper(X)) |
(49) |
proper(rnil) |
→ |
ok(rnil) |
(52) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
active(negrecip(X)) |
→ |
negrecip(active(X)) |
(14) |
proper(2ndsneg(X1,X2)) |
→ |
2ndsneg(proper(X1),proper(X2)) |
(56) |
negrecip(mark(X)) |
→ |
mark(negrecip(X)) |
(31) |
active(s(X)) |
→ |
s(active(X)) |
(12) |
pi(ok(X)) |
→ |
ok(pi(X)) |
(69) |
square(mark(X)) |
→ |
mark(square(X)) |
(45) |
active(pi(X)) |
→ |
pi(active(X)) |
(23) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(24) |
proper(pi(X)) |
→ |
pi(proper(X)) |
(57) |
active(square(X)) |
→ |
mark(times(X,X)) |
(11) |
active(times(0,Y)) |
→ |
mark(0) |
(9) |
active(posrecip(X)) |
→ |
posrecip(active(X)) |
(13) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(51) |
pi(mark(X)) |
→ |
mark(pi(X)) |
(40) |
2ndspos(ok(X1),ok(X2)) |
→ |
ok(2ndspos(X1,X2)) |
(67) |
proper(2ndspos(X1,X2)) |
→ |
2ndspos(proper(X1),proper(X2)) |
(55) |
proper(times(X1,X2)) |
→ |
times(proper(X1),proper(X2)) |
(59) |
active(pi(X)) |
→ |
mark(2ndspos(X,from(0))) |
(6) |
2ndsneg(mark(X1),X2) |
→ |
mark(2ndsneg(X1,X2)) |
(38) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(58) |
proper(posrecip(X)) |
→ |
posrecip(proper(X)) |
(48) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
proper(rcons(X1,X2)) |
→ |
rcons(proper(X1),proper(X2)) |
(53) |
proper(s(X)) |
→ |
s(proper(X)) |
(47) |
2ndspos(X1,mark(X2)) |
→ |
mark(2ndspos(X1,X2)) |
(37) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
proper(0) |
→ |
ok(0) |
(46) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
active(2ndspos(0,Z)) |
→ |
mark(rnil) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(179) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
active#(2ndsneg(X1,X2)) |
→ |
active#(X2) |
(185) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(134) |
active#(square(X)) |
→ |
active#(X) |
(123) |
active#(negrecip(X)) |
→ |
active#(X) |
(115) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(166) |
active#(s(X)) |
→ |
active#(X) |
(110) |
active#(2ndspos(X1,X2)) |
→ |
active#(X1) |
(108) |
active#(rcons(X1,X2)) |
→ |
active#(X2) |
(160) |
active#(from(X)) |
→ |
active#(X) |
(99) |
active#(2ndspos(X1,X2)) |
→ |
active#(X2) |
(155) |
active#(pi(X)) |
→ |
active#(X) |
(91) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(154) |
active#(2ndsneg(X1,X2)) |
→ |
active#(X1) |
(88) |
active#(rcons(X1,X2)) |
→ |
active#(X1) |
(87) |
active#(times(X1,X2)) |
→ |
active#(X1) |
(149) |
active#(times(X1,X2)) |
→ |
active#(X2) |
(80) |
active#(posrecip(X)) |
→ |
active#(X) |
(146) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
x1 + 18748 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
x1 + 1 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
3 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
x1 + x2 + 1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[2ndsneg(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[cons(x1, x2)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
x1 + 1 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(2ndsneg(X1,X2)) |
→ |
active#(X2) |
(185) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(134) |
active#(square(X)) |
→ |
active#(X) |
(123) |
active#(negrecip(X)) |
→ |
active#(X) |
(115) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(166) |
active#(s(X)) |
→ |
active#(X) |
(110) |
active#(2ndspos(X1,X2)) |
→ |
active#(X1) |
(108) |
active#(rcons(X1,X2)) |
→ |
active#(X2) |
(160) |
active#(from(X)) |
→ |
active#(X) |
(99) |
active#(2ndspos(X1,X2)) |
→ |
active#(X2) |
(155) |
active#(pi(X)) |
→ |
active#(X) |
(91) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(154) |
active#(2ndsneg(X1,X2)) |
→ |
active#(X1) |
(88) |
active#(rcons(X1,X2)) |
→ |
active#(X1) |
(87) |
active#(times(X1,X2)) |
→ |
active#(X1) |
(149) |
active#(times(X1,X2)) |
→ |
active#(X2) |
(80) |
active#(posrecip(X)) |
→ |
active#(X) |
(146) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
proper#(rcons(X1,X2)) |
→ |
proper#(X2) |
(135) |
proper#(2ndspos(X1,X2)) |
→ |
proper#(X1) |
(129) |
proper#(2ndsneg(X1,X2)) |
→ |
proper#(X1) |
(178) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(173) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(172) |
proper#(square(X)) |
→ |
proper#(X) |
(170) |
proper#(s(X)) |
→ |
proper#(X) |
(113) |
proper#(rcons(X1,X2)) |
→ |
proper#(X1) |
(111) |
proper#(from(X)) |
→ |
proper#(X) |
(109) |
proper#(2ndspos(X1,X2)) |
→ |
proper#(X2) |
(164) |
proper#(posrecip(X)) |
→ |
proper#(X) |
(163) |
proper#(2ndsneg(X1,X2)) |
→ |
proper#(X2) |
(105) |
proper#(negrecip(X)) |
→ |
proper#(X) |
(100) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(157) |
proper#(times(X1,X2)) |
→ |
proper#(X1) |
(156) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(86) |
proper#(pi(X)) |
→ |
proper#(X) |
(83) |
proper#(times(X1,X2)) |
→ |
proper#(X2) |
(78) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
x1 + 977 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
x1 + 1 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
2 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
x1 + x2 + 1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[2ndsneg(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
x1 + 0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
x1 + 1 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(rcons(X1,X2)) |
→ |
proper#(X2) |
(135) |
proper#(2ndspos(X1,X2)) |
→ |
proper#(X1) |
(129) |
proper#(2ndsneg(X1,X2)) |
→ |
proper#(X1) |
(178) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(173) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(172) |
proper#(square(X)) |
→ |
proper#(X) |
(170) |
proper#(s(X)) |
→ |
proper#(X) |
(113) |
proper#(rcons(X1,X2)) |
→ |
proper#(X1) |
(111) |
proper#(from(X)) |
→ |
proper#(X) |
(109) |
proper#(2ndspos(X1,X2)) |
→ |
proper#(X2) |
(164) |
proper#(posrecip(X)) |
→ |
proper#(X) |
(163) |
proper#(2ndsneg(X1,X2)) |
→ |
proper#(X2) |
(105) |
proper#(negrecip(X)) |
→ |
proper#(X) |
(100) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(157) |
proper#(times(X1,X2)) |
→ |
proper#(X1) |
(156) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(86) |
proper#(pi(X)) |
→ |
proper#(X) |
(83) |
proper#(times(X1,X2)) |
→ |
proper#(X2) |
(78) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
posrecip#(mark(X)) |
→ |
posrecip#(X) |
(103) |
posrecip#(ok(X)) |
→ |
posrecip#(X) |
(98) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
x1 + 0 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
x1 + 0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
x1 + x2 + 28652 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
56725 |
[mark(x1)] |
=
|
x1 + 0 |
[2ndsneg(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 0 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
x1 + 0 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
posrecip#(ok(X)) |
→ |
posrecip#(X) |
(98) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
times#(ok(X1),ok(X2)) |
→ |
times#(X1,X2) |
(180) |
times#(mark(X1),X2) |
→ |
times#(X1,X2) |
(177) |
times#(X1,mark(X2)) |
→ |
times#(X1,X2) |
(95) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
40482 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
x1 + 0 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
2 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[times#(x1, x2)] |
=
|
x2 + 0 |
[0] |
=
|
0 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 3448 |
[times(x1, x2)] |
=
|
x1 + x2 + 1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 1 |
[2ndsneg(x1, x2)] |
=
|
x2 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 0 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
x1 + 0 |
together with the usable
rules
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
times#(ok(X1),ok(X2)) |
→ |
times#(X1,X2) |
(180) |
times#(X1,mark(X2)) |
→ |
times#(X1,X2) |
(95) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
6th
component contains the
pair
2ndspos#(ok(X1),ok(X2)) |
→ |
2ndspos#(X1,X2) |
(151) |
2ndspos#(mark(X1),X2) |
→ |
2ndspos#(X1,X2) |
(84) |
2ndspos#(X1,mark(X2)) |
→ |
2ndspos#(X1,X2) |
(145) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
x1 + 1 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
x1 + 0 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
2 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
0 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
x1 + x2 + 1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 1 |
[2ndsneg(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 0 |
[2ndspos#(x1, x2)] |
=
|
x1 + x2 + 0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
x1 + 0 |
together with the usable
rules
times(X1,mark(X2)) |
→ |
mark(times(X1,X2)) |
(44) |
posrecip(mark(X)) |
→ |
mark(posrecip(X)) |
(30) |
posrecip(ok(X)) |
→ |
ok(posrecip(X)) |
(62) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
s(ok(X)) |
→ |
ok(s(X)) |
(61) |
times(ok(X1),ok(X2)) |
→ |
ok(times(X1,X2)) |
(71) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
from(ok(X)) |
→ |
ok(from(X)) |
(66) |
from(mark(X)) |
→ |
mark(from(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
times(mark(X1),X2) |
→ |
mark(times(X1,X2)) |
(43) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
2ndspos#(ok(X1),ok(X2)) |
→ |
2ndspos#(X1,X2) |
(151) |
2ndspos#(mark(X1),X2) |
→ |
2ndspos#(X1,X2) |
(84) |
2ndspos#(X1,mark(X2)) |
→ |
2ndspos#(X1,X2) |
(145) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(126) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(144) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
x2 + 0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
x1 + 2238 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
29509 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
10000 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
23236 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
2 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
7193 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 27541 |
[times(x1, x2)] |
=
|
23562 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
32610 |
[mark(x1)] |
=
|
23683 |
[2ndsneg(x1, x2)] |
=
|
12520 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + x2 + 23721 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
23682 |
together with the usable
rules
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(70) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(41) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(42) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(126) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
8th
component contains the
pair
pi#(ok(X)) |
→ |
pi#(X) |
(140) |
pi#(mark(X)) |
→ |
pi#(X) |
(158) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
28383 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
19213 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
64371 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 3 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
30384 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 14044 |
[2ndsneg(x1, x2)] |
=
|
x1 + 8683 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 17066 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
x1 + 0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
13964 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
pi#(ok(X)) |
→ |
pi#(X) |
(140) |
pi#(mark(X)) |
→ |
pi#(X) |
(158) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
9th
component contains the
pair
2ndsneg#(mark(X1),X2) |
→ |
2ndsneg#(X1,X2) |
(119) |
2ndsneg#(ok(X1),ok(X2)) |
→ |
2ndsneg#(X1,X2) |
(118) |
2ndsneg#(X1,mark(X2)) |
→ |
2ndsneg#(X1,X2) |
(81) |
1.1.9 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
19213 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
2 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 13021 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
30384 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 14044 |
[2ndsneg(x1, x2)] |
=
|
x1 + 2 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
x2 + 0 |
[posrecip(x1)] |
=
|
13964 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
2ndsneg#(ok(X1),ok(X2)) |
→ |
2ndsneg#(X1,X2) |
(118) |
2ndsneg#(X1,mark(X2)) |
→ |
2ndsneg#(X1,X2) |
(81) |
could be deleted.
1.1.9.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
10th
component contains the
pair
negrecip#(mark(X)) |
→ |
negrecip#(X) |
(139) |
negrecip#(ok(X)) |
→ |
negrecip#(X) |
(89) |
1.1.10 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
x1 + 0 |
[2ndspos(x1, x2)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
3868 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
16004 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
2 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
28473 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 14044 |
[2ndsneg(x1, x2)] |
=
|
x1 + 24813 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
13964 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
negrecip#(mark(X)) |
→ |
negrecip#(X) |
(139) |
negrecip#(ok(X)) |
→ |
negrecip#(X) |
(89) |
could be deleted.
1.1.10.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
11th
component contains the
pair
square#(mark(X)) |
→ |
square#(X) |
(94) |
square#(ok(X)) |
→ |
square#(X) |
(93) |
1.1.11 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
16209 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
64176 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
22166 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
x1 + 0 |
[pi(x1)] |
=
|
16415 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 22026 |
[times(x1, x2)] |
=
|
28473 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 14044 |
[2ndsneg(x1, x2)] |
=
|
x1 + 24813 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
13964 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
square#(mark(X)) |
→ |
square#(X) |
(94) |
square#(ok(X)) |
→ |
square#(X) |
(93) |
could be deleted.
1.1.11.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
12th
component contains the
pair
from#(ok(X)) |
→ |
from#(X) |
(183) |
from#(mark(X)) |
→ |
from#(X) |
(175) |
1.1.12 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 4330 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
12783 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
64176 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
22166 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
24632 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 15808 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 22026 |
[times(x1, x2)] |
=
|
28473 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 14044 |
[2ndsneg(x1, x2)] |
=
|
x1 + 24813 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
13964 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
from#(ok(X)) |
→ |
from#(X) |
(183) |
from#(mark(X)) |
→ |
from#(X) |
(175) |
could be deleted.
1.1.12.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
13th
component contains the
pair
rcons#(X1,mark(X2)) |
→ |
rcons#(X1,X2) |
(122) |
rcons#(mark(X1),X2) |
→ |
rcons#(X1,X2) |
(120) |
rcons#(ok(X1),ok(X2)) |
→ |
rcons#(X1,X2) |
(116) |
1.1.13 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
22166 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
19691 |
[rcons#(x1, x2)] |
=
|
x1 + 0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 24291 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 20541 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 1 |
[times(x1, x2)] |
=
|
28473 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 13779 |
[2ndsneg(x1, x2)] |
=
|
x1 + 24813 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
2526 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
rcons#(mark(X1),X2) |
→ |
rcons#(X1,X2) |
(120) |
rcons#(ok(X1),ok(X2)) |
→ |
rcons#(X1,X2) |
(116) |
could be deleted.
1.1.13.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
14th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(121) |
s#(mark(X)) |
→ |
s#(X) |
(75) |
1.1.14 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
644 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
27633 |
[plus#(x1, x2)] |
=
|
0 |
[square(x1)] |
=
|
30695 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
2 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 14989 |
[times(x1, x2)] |
=
|
28473 |
[s#(x1)] |
=
|
x1 + 0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 16161 |
[2ndsneg(x1, x2)] |
=
|
x1 + 24813 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
16714 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(121) |
s#(mark(X)) |
→ |
s#(X) |
(75) |
could be deleted.
1.1.14.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
15th
component contains the
pair
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(124) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(102) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(153) |
1.1.15 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[negrecip(x1)] |
=
|
2 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[negrecip#(x1)] |
=
|
0 |
[2ndspos(x1, x2)] |
=
|
31865 |
[top(x1)] |
=
|
0 |
[rnil] |
=
|
1 |
[plus#(x1, x2)] |
=
|
x1 + 0 |
[square(x1)] |
=
|
21654 |
[top#(x1)] |
=
|
0 |
[square#(x1)] |
=
|
0 |
[pi(x1)] |
=
|
32662 |
[rcons#(x1, x2)] |
=
|
0 |
[rcons(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[times#(x1, x2)] |
=
|
0 |
[0] |
=
|
1 |
[posrecip#(x1)] |
=
|
0 |
[from(x1)] |
=
|
x1 + 7885 |
[times(x1, x2)] |
=
|
28473 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 16161 |
[2ndsneg(x1, x2)] |
=
|
x1 + 2 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
x1 + 2 |
[2ndspos#(x1, x2)] |
=
|
0 |
[from#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
[pi#(x1)] |
=
|
0 |
[2ndsneg#(x1, x2)] |
=
|
0 |
[posrecip(x1)] |
=
|
12523 |
together with the usable
rules
rcons(X1,mark(X2)) |
→ |
mark(rcons(X1,X2)) |
(34) |
rcons(ok(X1),ok(X2)) |
→ |
ok(rcons(X1,X2)) |
(65) |
rcons(mark(X1),X2) |
→ |
mark(rcons(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(124) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(102) |
could be deleted.
1.1.15.1 Dependency Graph Processor
The dependency pairs are split into 1
component.